Abstract

Abstract This study exploits spatial anomalies in school funding policy in England to provide new evidence on the impact of resources on student achievement in urban areas. Anomalies arise because the funding allocated to Local Education Authorities (LEA) depends, through a funding formula, on the ‘additional educational needs’ of its population and prices in the district. However, the money each school receives from its LEA is not necessarily related to the school's own specific local conditions and constraints. This implies that neighbouring schools with similar intakes, operating in the same labour market, facing similar prices, but in different LEAs, can receive very different incomes. We find that these funding disparities give rise to sizeable differences in pupil attainment in national tests at the end of primary school, showing that school resources have an important role to play in improving educational attainment, especially for lower socio-economic groups. The design is geographical boundary discontinuity design which compares neighbouring schools, matched on a proxy for additional educational needs of its students (free school meal entitlement – FSM), in adjacent districts. The key identification requirement is one of conditional ignorability of the level of LEA grant, where conditioning is on geographical location of schools and their proportion of FSM children. 1. Introduction Improvement of the educational attainment of poor children is a top priority in many countries. This is a particular problem in countries, such as the United Kingdom and United States, where there are long tails in the bottom end of the adult distribution of basic literacy and numeracy skills, especially for the younger generation (OECD 1995, 2013). This bottom tail is heavily populated with people who have been disadvantaged since childhood, and many of these children live in inner-city urban areas.1 In the United Kingdom there is a substantial attainment gap at school entry between low-income pupils who are eligible to receive free school meals (FSM) and the rest, and this gap widens over time (National Equality Panel 2010).2 Recent academic work on addressing this gap, and raising achievements more generally, has turned attention towards institutional structures and incentives, such as greater school autonomy and competition. However, resources are an important part of educational policy and any lowering of real per pupil expenditure (as is happening currently in the United Kingdom) is extremely controversial. In England, districts and schools with poorer children receive more funding on the implicit assumption that more money helps. In this paper, we revisit this central question of whether simply allocating more money to schools results in higher achievement, where the context is urban areas (which tend to have a higher proportion of disadvantaged students). Our empirical analysis contributes to the literature by using a geographical boundary discontinuity design that focusses on the effects of explicit differences in the grants paid to neighbouring city schools districts in England. The design enables us to identify the effect of expenditure on pupil outcomes, despite the fact that central government funding to Local Education Authorities (LEAs) is explicitly compensatory. The research design is rooted in education funding policy anomalies in England which means that schools which are close together but in different education districts—LEAs—can get very different levels of core funding from central government. These funding differentials arise between schools that are otherwise very similar in their geographical location, catchment areas, student demographics, and the prices they face for their inputs.3 This discrepancy occurs because core funding is allocated to LEAs by central government according to: (a) the proportion of students from low-income families and those with English as a second language—“additional educational needs” (AEN); (b) an index (the “area cost adjustment”, ACA) computed for broader labour market areas that compensates the LEA for high labour costs; and (c) an adjustment for low population density to compensate for fixed costs in rural areas. However, the funding is not redistributed from LEAs to schools according to these rules. Therefore, two neighbouring schools in adjacent LEAs facing students with similar educational needs, staff wages, input prices, and other constraints can get different levels of funding simply because of the difference in the average educational needs of the LEA and the average market wages in the labour market area in which they are located. We exploit these features of England's educational system in a geographical boundary discontinuity design. This design matches schools according to school-level proxies for key characteristics that determine the grant their LEA receives: the proportion of children entitled to free meals, and geographical location. We then use the discontinuity in funding and student test scores between close-neighbouring primary schools in adjacent LEAs to estimate the causal effect of funding differentials on student outcomes. We use the “sharp” discontinuity in LEA-level average school grant that occurs at the boundary either directly in a reduced-form analysis, or as an instrument for school-level expenditure differences across the boundary. These designs are similar to classic “sharp” and “fuzzy” Regression Discontinuity Designs (Imbens and Lemieux 2008; Lee and Lemieux 2010). However, here we are matching schools both on spatial location and on student free meal entitlement, and exploiting a discontinuity in funding with respect to only one of these—geographical location. The identification condition is thus one of “conditional ignorability”: that is, conditional on the location of schools and the proportion of their students entitled to free meals, there are no confounders correlated with these LEA-level grant differences (see Keele et al. 2015 for further discussion in the context of geographical boundary discontinuity designs). In the education literature, similar techniques have often been used to look at the impact of school test scores on house prices as well as being used in other areas of economics and social sciences.4 The credibility of claims that any estimates represent causal parameters relies on ruling out alternative causal explanations, so we devote a lot of attention to this question. An important threat to designs of this type is sorting across the boundaries. In our context, the crucial concern is that there is sorting into schools on the basis of the grants paid to their LEAs. We argue that in our institutional setting and period of our study, these funding differences were hard to observe during the process of school choice, and expenditure provides a very weak or uninterpretable signal about school quality. These are not factors that parents consider. This is because of the complex compensatory way in which funds are allocated to schools that generally imply that on face value the “worst” schools receive the most money. In addition, we present an extensive range of balancing tests that demonstrate that our matched set of schools along the LEA boundaries are balanced in terms of observable covariates and other salient characteristics such as house price indices and mobility across LEA boundaries that would tend to indicate sorting. Additional tests show that many other factors, such as those related to local public finance, cannot explain our findings either. This investigation is important because of the policy concern to improve the educational attainment of disadvantaged students and in the light of the age-old debate about whether changes to school expenditure (at the levels typical in developed countries) really have any causal impact on outcomes. The causal effect of expenditure on outcomes is hard to identify, because money is often allocated to schools in ways that are correlated with pre-existing pupil advantages and disadvantages. There is also potential sorting of students into schools according to levels of expenditure. Studies that identify the effect in a convincing way are relatively few and economists have varying views on the interpretation of the literature. Hanushek (2008) argues that accumulated research says that there is currently no clear, systematic relationship between resources and student outcomes. Critics of this view suggest that many papers in this literature do not deal adequately with the endogeneity problem and/or have problems with data quality. There are indeed more papers with a strong methodology (using natural or real experiments) that show a positive effect from resource-related factors including Angrist and Lavy's (1999) study on the effect of class size in Israel; studies on the experimental Tennessee STAR class size reduction (Krueger 1999; Krueger and Whitmore 2001; Chetty et al. 2011); studies that have made use of student finance reforms (Guryan 2001; Van der Klaauw 2008a; Bénabou et al. 2009; Roy 2011; Hyman 2016; Jackson et al. 2016); and some of Hanushek's own work (Rivkin, Hanushek and Kain 2005). There have also been a couple of recent papers in England that have found modest effects of increased school resources (Jenkins et al 2006; Holmlund et al. 2010; Machin et al. 2010). However, there are studies with a strong methodology that find little or no effect of class size (e.g., Hoxby 2000; Cho et al. 2012), and thus the efficacy of general increases in school resources as a policy is still highly controversial amongst economists. A review of this literature is provided in Gibbons and McNally (2013). Our study is the first to our knowledge that applies the boundary discontinuity approach in order to provide causal estimates of the effect of school expenditure differentials. Because of the context of this study, our results apply to students who live in urban areas and are more likely to be disadvantaged. To preview our results, we show that schools that are well matched in terms of pupil characteristics, in different LEAs, but close to the boundaries, do receive different levels of core funding from central government, and that these differences in resources are associated with marked differentials in pupil performance. Our results imply that an additional £1,000 per student per year (a 30% increase relative to the mean) could raise achievement by around 30%–35% of a standard deviation. Furthermore, the effects are bigger in schools that have higher proportions of disadvantaged students. Our estimates are larger than those typically found in the literature for general resource increases, although comparable to the effects of class size reductions in the benchmark experimental study, the Tennessee STAR experiment (Schanzenback 2006). We suggest that our higher estimates arise because we focus on persistent cross-sectional differences in income and expenditure, whereas many previous studies have identified effects from short run time series variation within schools. Schools are likely to be able to adapt to short run fluctuations in funding and so may appear relatively unresponsive to resources in studies that exploit this type of variation. The remainder of our paper is structured as follows: we discuss the institutional structure of schools in England and how funding is allocated (Section 2); data (Section 3); empirical strategy (Section 4); regression results (Section 5); and discussion and conclusions (Section 6). 2. Institutional Context Underlying our Research Design Our research design uses district-level differences in per pupil funding in comparable close-neighbouring schools in a discontinuity-based design, to identify the causal effect of expenditure on student achievement. In this section, we present the relevant background information on the school system in England. The details of the data and research design are set out in Sections 3 and 4. The education syllabus is based on a National Curriculum and years of compulsory education are organised into four “Key Stages” (ending at the age of 7, 11, 14, and 16). At the end of primary school (end of Key Stage 2, age 11), all students in England undertake national tests in English, Maths, and Science. These are national tests that are externally set and marked and form the basis of School Performance Tables (or “league tables”). Our outcome variable will be these Key Stage 2 scores (ks2). For the period covered by our study (2004–2009), schooling was organised at the local level through LEAs.5 These LEAs often have the same geographical coverage as bodies that control other aspects of local government, such as social services, and civic amenities. In some cases, the geographical area of the LEA that is responsible for education encompasses a number of smaller authorities that are responsible for non–school-related services. There are 152 education-related LEAs in England,6 with an average of 23,500 primary school pupils, but with a lot of variation (standard deviation is 18,000). There are about 15,000 primary schools in England. Although the leaders in local government (councillors) are elected, they delegate all the day to day running of the services to appointed officials, and there is no equivalent to an elected “school board” as in the United States. Although LEAs have a number of roles in the provision of education, schools are largely self-governing. The LEA’s main functions in relation to primary schools are in building and maintaining schools, allocating funding, providing support services (e.g., for children with special needs), and acting in an advisory role to the head teacher regarding school performance and implementation of government initiatives. The LEA also appoints one or two representatives on to a school's governing body—a group of parents, teachers, and community representatives that provides governance to the school. Local Education Authorities typically also offer a number of administrative and management functions including training, personnel and financial services. The majority of pupils (67%) attend “Community schools” and for these schools, LEAs are also the statutory employer of school staff, owner of the buildings and the authority that manages student admissions. Most other state primary schools are faith schools (which have greater autonomy from the LEA). Student admissions are based on principles of parent/student choice rather than rigidly defined catchment areas, and parents can apply to any school. However, popular schools are oversubscribed and places are rationed according to various criteria such as priority for siblings, special educational needs (SEN), and proximity to the school. A Schools Admission Code dictates that student ability or family income cannot be used as a criterion. In our empirical analysis, we restrict attention to Community schools as they are more homogenous in their funding, governance, and admissions structure. Therefore, comparing adjacent Community schools in different LEAs makes it more likely we are comparing “like with like”. Around 85% of funding to schools comes from central government and general taxation. This funding is distributed to the LEAs, who then redistribute it to schools. Over most of the period relevant to this study, this core funding element was allocated to LEAs as a Formula Grant using a national formula. The key features of the primary school grant is that there is a basic allocation per pupil, with an allowance made for the labour market (“area cost adjustment”), LEAs with small schools in isolated areas (“sparsity”), and demographic “additional educational needs”. The AEN and ACA are the key components. Additional educational needs are based mainly on the proportion of families on Income Support, children with English as an Additional Language, and families with Working Families Tax Credits. We describe the ACA in detail later in this section. In addition to the this core Formula Grant, there is a variety of separate central government grants that are ring-fenced for specific purposes, such as to support national educational strategies for raising standards or ethnic minority achievement. These are passed on in full to schools by LEAs according to rules set by the Department for Education (e.g., according to school size or proportion of ethnic minorities). There have been many changes to the formulae and the various supplementary grants over time (as documented in West 2009). Notably, in 2006/2007 the Formula Grant and supplementary specific grants were combined into a “Dedicated School Grant”. However, the allocations across LEAs were based on the shares in previous years under the Formula Grant, so the basic allocations remain, implicitly, based on the same factors that determined allocations under the earlier formula (including the adjustments for area and educational need). Local Education Authorities use their own rules to allocate the core funding (other than the supplementary grants) to their schools, but the mechanisms have always been much less compensatory than those that allocate money to LEAs (West 2009). In many cases, the majority of funds were distributed equally on a per-pupil basis, with a small proportion targeted to the expected level of “special educational needs” of its pupils. This in itself implies that funding given to a school has a component that is more closely related to the demographics in the LEA as a whole than it is to the demographics of the school itself, giving rise to potential funding gaps between comparable schools in different, less comparable LEAs. We use LEA-level, rather than school-level funding variables to identify the effects of resources, so the exact allocation within LEAs is not crucial to our analysis. When the funding gets to schools, it is for the school to decide how to use it, although the bulk of expenditure is on teacher pay. The broad allocation of spending is as follows: 60% on teachers; 20% on support staff or other staff; 6% on building and maintenance; 5% on learning resources/IT and 8% on a residual category. This has changed little over time (Holmlund et al. 2010). An important element of our research design is the ACA in the national funding formula that is intended to compensate LEAs for differences in costs. This reflects two kinds of difference between areas in local costs: differences in labour costs (i.e., the main factor) and differences in business tax rates paid on local authority premises. The “labour cost adjustment” is based on differences in mean wages (private and public sector) between areas, regression-adjusted for worker and job characteristics.7 The labour cost adjustments are intended to compensate for market wage differentials, which in turn are compensating workers for differences in housing costs and amenities (Roback 1982; Moretti 2013). Public sector employment accounted for around 19% of total employment in England at the end of the period we study, so the estimated ACAs are heavily weighted towards private sector pay differentials. The underlying rationale in applying these ACAs to public finance allocations is that local public services have to compete for staff with other employers and therefore authorities in areas with high private sector wages can face higher staffing costs.8 However, teachers and other school staff and LEA employees are public sector workers, and in practice public sector pay is far less variable geographically than private sector pay (Bell et al. 2007; Emmerson and Jin 2012). This is because the public sector has greater union coverage, and unions typically work against the introduction of region-specific pay scales, which they see as unfairly disadvantaging workers in low pay areas. Therefore, this extra funding to LEAs does not necessarily get passed on to staff or teachers in higher basic pay, as they get paid according to national pay scales with very limited regional variation (see below). However, the money can implicitly be used in other ways, to increase staff numbers, provide extra material resources or support services, or could potentially be used to pay recruitment or performance incentives to teaching staff. This discrepancy between the way the ACA is estimated, and the reality of the pay environment faced by schools, is one reason why the ACA can lead to school funding differentials in real terms. A second reason is that the ACAs are defined for 54 subregional geographical units that are aggregates of the 152 LEAs, so neighbouring LEAs can receive different levels of per-pupil funding simply because they have been allocated to different ACA regions. There are, however, some cases in which teachers in adjacent LEAs will receive different nominal pay automatically. Teacher national pay scales in England have London allowances, which vary according to whether the school in which they work is in one of four pay zones: Inner London, Outer London, London Fringe, or the Rest of England. These pay zones do not coincide with the 54 geographical areas on which the ACA is based, but they are geographical aggregates of contiguous LEAs. Hence, there will be some cases where teachers working on different sides of an LEA boundary get different nominal pay through the national pay scales, and hence higher real wages, assuming that teachers working in neighbouring schools will not genuinely face higher housing or consumption costs. To the extent that these higher real wages induce improvements in teacher quality and hence student performance, it is appropriate to include them within our expenditure measures. In practice, other evidence suggests that these teacher pay differentials do not affect student performance (Greaves and Sibieta 2014)9, but we will in any case test for the robustness of our results to controls for these London teacher pay allowance zones. A related concern would be if neighbouring schools genuinely faced different prices for their other labour inputs (e.g., bought in private sector services). Similarly, there would be a concern if any differentials in nominal wages paid to teachers simply compensated them for differential commuting, goods or housing costs and resulted in no improvement in teacher quality through selection or effort. Our underlying assumption then is that market prices and wages do not vary between close-neighbouring schools, and any differential in expenditure and teacher pay induced by institutional funding rules represent real expenditure and wage differences. This assumption seems relatively uncontroversial, given that schools we compare in our sample are on average only 1.4 km apart. Average commuting distance in metropolitan areas in Britain is around 11 km (figures taken from the National Travel Survey 2010), so teachers and other workers are unlikely, on average, to view either one of two schools 1.4 km apart as preferable to the other on the basis of distance alone. The official labour market definitions for the United Kingdom are Travel to Work Areas (TTWAs), have an average land area of 1000 km2 that is obviously large compared to the scales at which we are working.10 London is a single TTWA with land area of 2700 km2. 3. Data Our study is based on the National Pupil Database (NPD, a census of all students in state schools) between academic years 2003/2004 and 2008/2009. The data set contains information on the national ks2 test in English, Maths, and Science, taken in May. There is no grade repetition in the English system so all pupils are in the same year group when they take these tests. We use the average score across these subjects as our main outcome variable. We do not have full information on funding before 2002/2003 and wish to include time lagged funding data so we restrict attention to 2003/2004 onwards. We do not use years beyond 2009 because institutional changes occurred thereafter. The NPD also has information on the prior test scores (ks1) of each student at age 7 in Reading, Writing, and Maths. Demographic information included in the data set relates to gender, ethnicity; whether English is his/her first language; whether the pupil is known to be eligible for FSM (an important indicator of socioeconomic disadvantage). Geographic information on the pupil's home residence is also available at Census “Output Area” (i.e., small geographic clusters of households).11 This can be linked Census data from 2001 and an index of income deprivation: the IDACI index (i.e., Income Deprivation Affecting Children: an index based on the proportion of children under the age of 16 who live in low-income households). Additional data at school level come from the Annual School Census (such as pupil numbers; the proportion of pupils eligible for FSM in the School). Information on school expenditure and income is taken from the “Consistent Financial Reporting” (CFR) data set, which contains detailed information on school expenditure and income sources in the financial years. Financial years in England begin in April, so the expenditure in a given financial year relates (approximately) to the annual period leading up to the Key Stage tests in May. All the expenditure and income variables we use exclude capital expenditure for building work, which is funded separately. We have information on national funding formula for LEAs in each year, including how funding is allocated on the basis of AEN, ACA, and sparsity. To set up these data for our empirical analysis, we carry out a number of data manipulations using a Geographical Information System, computing distances between each school and its nearest neighbours based on the school postcode coordinates, distances to Local Education Authority boundaries. We also derive a subset of LEA boundaries that do not coincide with geographical features (major roads, motorways, railways) using feature data from the Ordnance Survey (these geographical data were obtained from the geographical data service at www.edina.ac.uk). 4. Empirical Strategy 4.1. Introductory Outline Our aim is to estimate the impact of school resources on student achievement. The first-order problem is that resources are systematically allocated to schools to compensate for student disadvantages. To do so, we compare performance in geographically neighbouring, Community schools close to LEA boundaries, in adjacent LEAs. These LEAs receive different levels of compensatory grant from central government, though the schools we compare face similar demographic and labour market conditions. To ensure this, we match schools not only on geographical location as in standard geographical boundary discontinuity design, but also on an observable school-level proxy for a key element in the formula that determines central government grants to LEAs—the proportion of children eligible for FSM. This matched, geographical regression discontinuity design is broadly similar to that proposed by Keele et al. (2015), who describe the corresponding identifying condition as “conditional geographic treatment ignorability”. In the next sections, we set out the underlying structure of relationships between funding, socioeconomic characteristics and school performance that justifies this identification strategy, and set out the design in more detail. 4.2. Eliminating Biases from Compensatory School-funding Mechanisms We compare test scores and expenditure in pairs of schools s and s΄ in adjacent LEAs k and k΄ in a regression specification with the structure: \begin{equation} (ks{2_{sk}} - ks{2_{s'k'}}) = \beta ({\mathit{expend}}_{sk} - {\mathit{expend}_{s'k'}}) + \lambda ({v_s} - {v_{s'}}) + ({\varepsilon _s} - {\varepsilon _{s'}}). \end{equation} (1) Here, ks2s is school-mean student Key Stage 2 (age 11) test scores (an average across three subjects: Maths, Science, and English) and expendsk is a measure of per-student, current expenditure in school s in the years leading up to the age 11 test. We estimate student-level regressions using data from multiple years, but for exposition we first discuss the design using school-level expressions.12 Variable vs is a partially observed index of specific school/demographic characteristics which are, at LEA level, salient in determining the compensatory funding that the LEA receives from central government.13 Component εs represents other school-level unobservables, potentially correlated with vs. As discussed in Section 2, the central government grant allocated to an LEA in a given year is based on a national per-pupil baseline, plus components compensating for the LEA’s low-income population (AEN), labour input prices (ACA), and residential density (“sparsity”). So, richer, high-wage, dense LEAs on average will get less money per pupil than poorer, low-wage, sparse LEAs. We expect vs to vary smoothly across space, relative to school performance, because it represents local labour market conditions and sorting on amenities and housing that are unrelated to school provision. We can think of vs as having two spatial components: vk represents the LEA-mean characteristics that directly determine LEA funding that is, grantk = vk;14 school-level components $${\tilde{v}_s}$$ represents the deviations from this mean such that $${v_s} = {v_k} + {\tilde{v}_s}$$. Each LEA redistributes the grant to schools according to its own rules each school has its own idiosyncratic sources of income, implying the following “first stage” equation:15 \begin{eqnarray} ({\mathit{expend}}_{sk} - {\mathit{expend}}_{s'k'}) &=& \gamma ({\mathit{grant}}_k - {\mathit{grant}}_{k'}) + ({\varsigma _s} - {\varsigma _{s'}})\nonumber \\ &=& \gamma ({v_k} - {v_{k'}}) + ({\varsigma _s} - {\varsigma _{s'}}). \end{eqnarray} (2) Clearly, school expenditure is explicitly endogenous in equation (1) as is the LEA grant in the reduced form obtained by substituting equation (2) into (1). Both LEA grant and school expenditure depend directly on vk: the index of LEA-mean low income, labour costs, and sparsity. A second endogeneity concern from the funding mechanism is that school-specific expenditure (ςs) is determined by other factors (εs) that affect achievement in equation (1). This correlation arises because there are school-specific characteristics affecting school-specific funding, conditional on the grant (grantk) provided to the LEA. The reasons are as follows: (a) some LEAs reallocate their grant across schools to compensate for intake demographics or to target low achievement, but not in a way that follows the central government funding formula;16 (b) LEAs provide additional resources to schools for students with disabilities and SEN; (c) schools receive some additional grants direct from central government to target national strategies and ethnic underachievement, which depend on a school's student intake characteristics; (d) a school can raise its own resources through fundraising events, applying for specific grants and charitable donations and its success in doing so will depend on the needs of its students and the motivation and skills of the leadership team. Our research design tackles these two issues in two steps. The first step is to eliminate the school-level factors (vs) that are related by construction to the LEA-grant funding formula. Note that for pairs of schools s and s΄, which are in different LEAs k and k΄ but which have similar school-level characteristics vs, it must be the case that $${v_k} + {\tilde{v}_s} \approx {v_{k'}} + {\tilde{v}_{s'}}$$, but need not be the case that vk = vk΄.17 This implies that for a pair of schools, s and s’, which are exactly matched on vs but are in different LEAs: \begin{equation} (ks{2_{sk}} - ks{2_{s'k'}}) = \beta ({\mathit{expend}}_{sk} - {\mathit{expend}}_{s'k'}) + ({\varepsilon _s} - {\varepsilon _{s'}}). \end{equation} (3) Thus, two LEAs containing these schools explicitly receive different levels of funding because they differ in mean characteristics vk (equation 2), but these differences are eliminated at school level by differencing between schools that are matched in terms of vs (equation 3). This differencing also eliminates other unobserved school components in εs that are correlated with vs. The credibility of this step depends on making any residual differences in the AEN, labour market, and sparsity of matched schools (vs − vs΄) small, random, and uncorrelated with the differences in the LEA grants(grantk − grantk΄), satisfying the “conditional geographic ignorability” assumption. We describe this matching of schools across LEA boundaries in Section 4.3. The second step is to eliminate biases from the correlation between school-specific characteristics and resources (E[(εs − εs΄)(ςs − ςs΄)] ≠ 0) by exploiting exogenous variation in the difference in the LEA grants(grantk − grantk΄). We do this either in a reduced form or instrumental variables (IV) specification implied by first-stage equations (2) and second-stage equation (3). As an alternative to using the LEA grant as an instrument, we also use the ACA element of the index (vk − vk΄) that is designed to account for differences in wages (regression adjusted for demographics) between the two LEAs. An advantage of using the ACA as an instrument is that market wage levels vary more smoothly over space than demographics, so it is more likely that neighbouring schools s and s΄are matched in terms of their labour market conditions (even though their LEAs as a whole are not).18 In other words, the assumption of conditional geographic treatment ignorability is more likely to hold. The remaining sources of endogeneity in equations (2) and (3) are correlation between the cross–LEA-boundary differences in school unobservables and the cross–LEA-boundary difference in LEA grants (i.e., E[(εs − εs΄)(grants − grants΄)] ≠ 0). This correlation could arise if there is sorting of students into schools across LEA boundaries based on the LEA grant (or ACA index), or if there are inputs other than school expenditure that are correlated with the LEA grant (or ACA index). We discuss these threats in Section 4.4. 4.3. Matching of Schools by Location and Salient Demographics If we fully observed the formula used by central government to allocate grants to LEAs, and observed the variables (vs) in this formula at school level, then we could match schools in different LEAs based exactly on these variables and implement the differencing in equations (2) and (3) on these matched pairs. In practice, this approach is infeasible. We do not observe the proportion with AEN at school level, defined exactly as in the formula. Nor can we observe at school level the relevant labour market characteristics used to construct the ACA index or sparsity index. An alternative, purely geographical discontinuity design following traditional Regression Discontinuity Design (RDD) practice would compare neighbouring schools on opposite sides of LEA boundaries and control for smooth variation in unobservables across space using a parametric or semiparametric function of distance to boundary. In our case, this is unlikely to eliminate differences in vs. The units of observation (schools) are geographically fairly sparse, there are multiple discontinuities (boundaries), the assignment variable is multidimensional (two-dimensional geographical space), and the threshold is a complex function of the assignment variable that defines the geographical boundaries. Moreover, the appropriate shape of the assignment variable control function differs from boundary to boundary. As noted by Keele et al. (2015), simple distance-to-boundary controls are unlikely to suffice. Instead, we use a combination of these approaches. We match schools in different LEAs based on two things: (a) the proportion of students who can claim FSM, which is an observable school-level proxy for the proportion on income support benefits that enters the AEN index at LEA level; (b) geographical location, which serves as a control for labour market conditions in the ACA and other components of the funding formula. Similar matching methods have been proposed by Keele et al. (2015) to address the difficulty of controlling for spatial trends in geographic discontinuity designs. In our case, matching on FSM is not ad hoc but informed by the crucial role of the AEN index in determining LEA grants. Specifically, we match schools to neighbouring schools in the boundary data set that are within +/– 10 percentiles in the distribution of the school proportion of FSM (we explore sensitivity to this matching threshold in the empirics). Appendix A further illustrates the justification and consequences of this FSM-matching procedure. When matching schools based on spatial location and controlling for unobserved smooth spatial trends, we follow the standard practice of restricting the sample to areas close to boundaries and controlling for distance to boundary trends in various ways, including polynomials, linear trends, boundary-specific linear trends, and locally weighted regressions where we apply higher weights to more closely spaced schools. 4.4. The Threat from Sorting and Other LEA Inputs Even with the above cross-boundary differencing and instrumental variable strategy, we are left with a threat from potential sorting into schools, and unobserved factors at LEA level that might be correlated with the LEA grant or ACA instruments. As is well known (Imbens and Lemieux 2008), discontinuity designs can fail if agents can precisely and strategically manipulate which side of the boundary they are on. Whereas schools are under LEA control and cannot relocate to a different LEA, families can choose on which side of the boundary to live and go to school. This problem of “sorting” across geographical boundaries is pervasive in geographical discontinuity designs applied to house prices, where a standard solution or robustness check is just to control for observables (Bayer, Ferreira and McMillan 2007; Gibbons, Machin and Silva 2013). Given our institutional context, this problem is not necessarily very severe. The threat here is from sorting by ability on the LEA grant or ACA index used as instruments, not on school-specific components of expenditure that are uncorrelated with the LEA grant. Moreover, school choice is generally not made on the basis of expenditure. Burgess et al. (2009) find that “closest or nearness to home” and “general good impression” are the most frequently citied reasons for choosing schools. Publicly available “league tables” provide one information source for parents, but these focus mainly on headline test scores. Information on school expenditure was publicly unavailable and was only introduced into the league table information in 2010 after the period of analysis. Information on the LEA grant was not published and would require considerable research to disentangle. Moreover, even if school expenditure or the LEA grant is observed, it is impossible for parents to interpret this as a signal of school quality given the compensatory nature of funding. Higher funding and expenditure signal that the school faces significant challenges. In addition, slightly higher levels of mean LEA grant does not necessarily correspond to higher levels of school expenditure or school performance for individual schools close to the boundary, because they can be offset by the idiosyncratic sources of variation discussed in Section 4.2. Although it is true we will find that on average across all boundaries, a higher LEA grant is associated with higher school expenditure and improved outcomes (this is the first stage, and main finding in our regressions); this is not true across each and every boundary, so a parent trying to choose between two schools in adjacent LEAs would have no way to easily understand the implications of higher school expenditure in particular schools. In other words, the exogenous variation in funding that we use for identification is either not observable by parents (because it is not completely revealed in school expenditures or outcomes) or not informative (because it is uninterpretable). Note that as well as controlling for a principal determinant of LEA grants at school level, matching on FSM proportions as described in Section 4.3 has the advantage of controlling for other forms of selection into high- and low-funded schools, where this selection is on observables that are correlated with FSM. We present extensive tests to show that our matched schools are balanced in terms of observable school and student characteristics and that our findings are not driven by sorting on these characteristics. A final concern is that the LEA grant might correlate with other LEA-specific nonschool inputs such as social services that might affect children's education. Although we cannot rule this out a priori, we present further discussion and a number of tests in Section 5 which demonstrate that other LEA-specific factors are not credible as alternative explanations for the relationship between school resources and student performance that we uncover in the empirical analysis. 4.5. Data Structure and Estimation In practice, we estimate equations (2) and (3) by running the regressions on student-level data with the paired s-s’ school observations stacked in a panel structure, with school-s-by-year fixed effects in order to obtain estimates based on differences in funding within these school pairs in each year. In other words, we control for school-pair-by-year fixed effects (in robustness checks, we generalise to match each school s to two or more of its closest matched neighbours s’, s’’, s’’’ and control for school-group-by-year fixed effects). This arrangement with repeated schools generates a complex data and error structure. To make our standard errors robust to this, and to other forms of spatial and serial autocorrelation along and across the LEA boundaries, we “cluster” the standard errors on LEA boundary groups. Another important point to note is that this research design necessarily creates a selected subsample of schools and students, so there is a potential sacrifice of external validity for the sake of internal validity. The schools in these boundary subsamples are primarily urban (given the greater density of schools and LEA boundaries within urban areas) and the students from more educationally disadvantaged background. 4.6. Interpretation as Sharp and Fuzzy RDD Note that the reduced form regression is analogous to a sharp Regression Discontinuity Design (Imbens and Lemieux 2008; Lee and Lemieux 2010) in the sense that all schools on one side of a given LEA boundary are assigned the same constant treatment and all schools on the other side are assigned a different constant treatment (the treatments are the LEA-specific grant on each side, whereas in the classic RDD these constant values would be 0 and 1). The IV approach is analogous to a fuzzy RDD in that the treatment takes many values (the school-specific expenditure) on each side of a given boundary (in the paradigmatic fuzzy RDD these would be a mixture of 0s and 1s). In this geographical regression discontinuity design, there are of course also many different boundaries and therefore many different discontinuities (152 in our main estimates). Therefore, estimation is based not just on the existence of a jump in the LEA grant from one side to the other, but from the magnitude of this jump and its correlation with the jump in test score outcomes across these different boundaries. 5. Results 5.1. Description of the Sample Table 1 shows descriptive statistics. The left-hand side of the top panel shows figures for the full national student sample for comparison purposes. The right-hand side of the top panel shows figures for the students in pairs of schools that are matched across LEA boundaries as described in Section 4. Figure 1 maps the schools in this boundary subsample. As we have discussed, our research design brings the focus on urban schools because of the greater density of boundaries and schools in urban areas. Schools that are close to boundaries either in or on the periphery of London account for 60% of the sample (as compared to 14% in the population overall). Figure 1. View largeDownload slide Distribution of schools in the school-pair boundary subsample. Figure 1. View largeDownload slide Distribution of schools in the school-pair boundary subsample. Table 1. Descriptive statistics. Expenditures are pounds per pupil. Full data set Matched school pair boundary subsample Mean s.d. Mean s.d. Student data set Age-11 total score 0 1.0 –0.030 0.990 School total expenditure (mean annual) £3,312.5 £636.3 £3,763.0 £844.3 Income from LEA grants (mean annual) £2,633.7 £264.5 £2,922.7 £433.9 ACA index 1.042 0.064 1.123 0.100 Distance to LEA boundary (metres) 492.6 550.2 Distance between paired schools 1,392.0 430.0 Boys 0.509 – 0.506 – FSM 0.165 – 0.294 – Age in months (within year) 5.471 3.484 5.532 3.486 English first language 0.881 – 0.669 – White British 0.820 – 0.549 – Student observations 3,311,712 379,194 Schools 15,308 840 LEAs 150 103 School-pair-by-year groups – 4,540 LEA-pair-by-year groups – 593 Mean difference (absolute values) s.d. 95th percentile Cross-boundary funding differences in school-by-year level data set Total school expenditure per pupil per year £505.9 £442.6 £1,285.4 Cross-boundary funding differences in LEA-by-year level data set Mean grant from LEA £237.7 £326.4 £782.1 ACA index 0.026 0.0581 0.164 Full data set Matched school pair boundary subsample Mean s.d. Mean s.d. Student data set Age-11 total score 0 1.0 –0.030 0.990 School total expenditure (mean annual) £3,312.5 £636.3 £3,763.0 £844.3 Income from LEA grants (mean annual) £2,633.7 £264.5 £2,922.7 £433.9 ACA index 1.042 0.064 1.123 0.100 Distance to LEA boundary (metres) 492.6 550.2 Distance between paired schools 1,392.0 430.0 Boys 0.509 – 0.506 – FSM 0.165 – 0.294 – Age in months (within year) 5.471 3.484 5.532 3.486 English first language 0.881 – 0.669 – White British 0.820 – 0.549 – Student observations 3,311,712 379,194 Schools 15,308 840 LEAs 150 103 School-pair-by-year groups – 4,540 LEA-pair-by-year groups – 593 Mean difference (absolute values) s.d. 95th percentile Cross-boundary funding differences in school-by-year level data set Total school expenditure per pupil per year £505.9 £442.6 £1,285.4 Cross-boundary funding differences in LEA-by-year level data set Mean grant from LEA £237.7 £326.4 £782.1 ACA index 0.026 0.0581 0.164 Notes: Table reports means and standard deviations. Top panel summarises student-level data. Bottom panel summarises funding differentials in data set collapsed to LEA-k by LEA-k’ cell means. Data covers years 2003/2004 to 2008/2009. Expenditure and income in 2009 prices (deflated by Consumer Price Index). View Large Table 1. Descriptive statistics. Expenditures are pounds per pupil. Full data set Matched school pair boundary subsample Mean s.d. Mean s.d. Student data set Age-11 total score 0 1.0 –0.030 0.990 School total expenditure (mean annual) £3,312.5 £636.3 £3,763.0 £844.3 Income from LEA grants (mean annual) £2,633.7 £264.5 £2,922.7 £433.9 ACA index 1.042 0.064 1.123 0.100 Distance to LEA boundary (metres) 492.6 550.2 Distance between paired schools 1,392.0 430.0 Boys 0.509 – 0.506 – FSM 0.165 – 0.294 – Age in months (within year) 5.471 3.484 5.532 3.486 English first language 0.881 – 0.669 – White British 0.820 – 0.549 – Student observations 3,311,712 379,194 Schools 15,308 840 LEAs 150 103 School-pair-by-year groups – 4,540 LEA-pair-by-year groups – 593 Mean difference (absolute values) s.d. 95th percentile Cross-boundary funding differences in school-by-year level data set Total school expenditure per pupil per year £505.9 £442.6 £1,285.4 Cross-boundary funding differences in LEA-by-year level data set Mean grant from LEA £237.7 £326.4 £782.1 ACA index 0.026 0.0581 0.164 Full data set Matched school pair boundary subsample Mean s.d. Mean s.d. Student data set Age-11 total score 0 1.0 –0.030 0.990 School total expenditure (mean annual) £3,312.5 £636.3 £3,763.0 £844.3 Income from LEA grants (mean annual) £2,633.7 £264.5 £2,922.7 £433.9 ACA index 1.042 0.064 1.123 0.100 Distance to LEA boundary (metres) 492.6 550.2 Distance between paired schools 1,392.0 430.0 Boys 0.509 – 0.506 – FSM 0.165 – 0.294 – Age in months (within year) 5.471 3.484 5.532 3.486 English first language 0.881 – 0.669 – White British 0.820 – 0.549 – Student observations 3,311,712 379,194 Schools 15,308 840 LEAs 150 103 School-pair-by-year groups – 4,540 LEA-pair-by-year groups – 593 Mean difference (absolute values) s.d. 95th percentile Cross-boundary funding differences in school-by-year level data set Total school expenditure per pupil per year £505.9 £442.6 £1,285.4 Cross-boundary funding differences in LEA-by-year level data set Mean grant from LEA £237.7 £326.4 £782.1 ACA index 0.026 0.0581 0.164 Notes: Table reports means and standard deviations. Top panel summarises student-level data. Bottom panel summarises funding differentials in data set collapsed to LEA-k by LEA-k’ cell means. Data covers years 2003/2004 to 2008/2009. Expenditure and income in 2009 prices (deflated by Consumer Price Index). View Large The first row in Table 1 shows the descriptives for ks2 test scores, which have zero mean and unit standard deviation in the full sample by construction. In the boundary, subsample mean scores are slightly lower. The second row shows school expenditures, averaged over the maximum number of years for which data are available between the ks1 and ks2 tests (rising from 2 years in 2002 to 3 years in 2003 and 4 years from 2004 onwards). This expenditure figure is in 2009 prices and is intended to estimate the average expenditure during each year of the Key Stage 2 period (between ages 7/8 and 10/11). Note that within schools the correlation in spending across years is very high (r around 0.8) and our analysis is cross-sectional, so whether we consider 1 year or multiyear means is largely irrelevant. The boundary subset of schools has higher levels of per-student spending than the national average (£3,763 compared to £3,313 per pupil on average at 2009 prices). The third row shows the LEA grant per pupil, which will provide the first source of identification in our empirical analysis in what follows. The fourth row shows the ACA index. The schools in our boundary sample have higher levels of LEA grant (£2,923 per pupil per year compared to £2,634 nationally) and a higher ACA index (12% compared to 4%). Note that this core LEA grant is 78% of school income on average in our boundary schools. As discussed in Sections 2 and 4, the remaining income comes from various sources which are school-specific. There is an LEA budget targeted at the expected SEN in the schools, which amounts to around 8.5% of school income in our boundary sample. There are various specific grants from central government for specific national strategies amounting to around 9.5% (such as the “Standards Fund”, 6% and ethnic minority achievement, 2%), and these are allocated by LEAs to schools according to national rules based on the school intake characteristics. Other school-varying sources make up about 4%, including catering, other services, income from school trips, voluntary donations and insurance claims. Note that the standard deviation of school expenditure is about twice that for the core LEA grant (£844 vs £434 in our boundary sample), showing that expenditure is much more variable across schools than is the core LEA grant. This higher variability in school total expenditure is due to the fact that these additional income sources are much more dependent on the school's specific socioeconomic characteristics. Children in the boundary schools are more likely to be on FSM, less likely to speak English as a first language, and less likely to be White British, reflecting their urban locations. The table also summarises the distances between our matched boundary schools. The schools are close to each other, being less than 1.4 km apart on average (about a 15 minute walk) and less than 500 metres from the LEA boundary. The lower panel of Table 1 shows the differences in school expenditure, LEA grant, and ACA index across the LEA boundaries, based on the boundary subsample of schools and students (aggregated to LEA-by-year cells). We report the mean of the absolute value of the deviation, the standard deviation and the 95th percentile. The mean average absolute cross-boundary difference in school expenditure is just over £500 per pupil per year, the standard deviation is £443 and the 95th percentile is just under £1,300. These differences are substantial, representing respectively 13%, 11%, and 34% of the average per pupil expenditure in the boundary schools. The differences in the LEA grant are similarly large, with a mean of £238, standard deviation of £326, and 95th percentile equal to £782, corresponding to 8%, 11%, and 27% of the average per pupil LEA grant. Note again that around half of the absolute difference in school expenditure across the boundaries is due to the LEA grant differential and around half is due to idiosyncratic school-specific differences in income, arising from the various school-varying income sources discussed previously. If we regress cross-boundary expenditure differences on cross-boundary LEA mean grant differences—the first stage regressions in our subsequent IV estimations—the R-squared is around 15%. This point is important for our subsequent discussion of sorting on LEA grants: schools in LEAs with high central government grants often do not have higher expenditure. There is also marked variation in the ACA index, with an average differential of 2.6%, a standard deviation of 5.8%, and a 95th percentile of 16.4%. These figures include the cases where there is zero differential in the ACA index between the adjacent LEAs in our boundary subsample. Note these descriptive statistics for the boundary subsample related to 216 LEA-pair-year observations with nonzero ACA index differences, corresponding to combinations of 29 different LEAs. The total number of LEAs represented in our boundary sample is 103 (out of 150 in England). Figure 2 presents this information in another way, and shows the distribution of the between-LEA funding differentials in the school-by-year level data. This top figure shows the variation in the LEA grant between schools in adjacent LEAs, and which we use to estimate the sharp discontinuity regressions in the next section, and which we use as the first instrument for school expenditure differences. The bottom panel shows the variation in the ACA index between schools in adjacent LEAs; this variable provides the second instrument for school expenditure differences. The figures show that there is substantial variation in both of these funding variables. Figure 2. View largeDownload slide (a) LEA grant residuals across boundaries (£000s). (b) ACA index residuals across boundaries. Figures show histograms of residuals from regressions of funding variables on LEA-k fixed effects and school distance to boundary polynomial, using student data set collapsed to LEA-k by LEA-k’ cell means. Figure 2. View largeDownload slide (a) LEA grant residuals across boundaries (£000s). (b) ACA index residuals across boundaries. Figures show histograms of residuals from regressions of funding variables on LEA-k fixed effects and school distance to boundary polynomial, using student data set collapsed to LEA-k by LEA-k’ cell means. 5.2. Evaluating the Identification Strategy Part 1: Balancing Tests The validity of our research design relies on the cross-boundary LEA grant and ACA differentials in our data set of matched boundary schools being uncorrelated with potential confounders. We therefore first present a series of “balancing” tests to show to what extent our process of matching on distance and FSM is successful, and the instruments used in the regression analysis are uncorrelated with differences in student characteristics across LEA boundaries (more detail on how FSM matching works and affects this balancing is given in Appendix A). These tests are highly relevant, assuming that selection on observables provides some guide to selection on unobservables too (Altonji et al. 2005). We do this in Table 2. The table presents coefficients from regressions of various student, school, and residential socioeconomic variables on our identifying instruments. Each row represents a different regression and shows the dependent variable (e.g., expenditure per pupil in row 1). Columns (1), (3), and (5) show coefficients on the LEA grant per pupil. Columns (2), (4), and (6) show coefficients on the ACA index. Columns (1) and (2) present the results using our preferred strategy that matches each school to its nearest school in an adjacent LEA amongst the set with an FSM proportion within +/–10 deciles, and controls for distance-to-boundary polynomials. Columns (3) and (4) show what happens when we remove this FSM matching constraint. Specifications in columns (5) and (6) control for FSM decile dummies in the sample of boundary schools, but with no matching by distance, cross-boundary differencing or controls for spatial trends. In all these regressions, the data are aggregated to school-by-year cells. The instruments are standardised (dividing by their standard deviation in the sample) so that the coefficients on the ACA index and LEA grant per pupil can be compared. Table 2. “Balancing” tests on boundary schools. (1) (2) (3) (4) (5) (6) Matched on location and FSM Matched on location only Controlling for FSM only Std LEA grant Std ACA Std LEA grant Std ACA Std LEA grant Std ACA Expend. per pupil 0.425*** 0.280*** 0.477*** 0.419*** 0.641*** 0.449*** (0.065) (0.075) (0.083) (0.061) (0.017) (0.030) Key stage 2 0.133** 0.064* 0.034 –0.022 0.096*** 0.085*** (0.045) (0.032) (0.067) (0.066) (0.014) (0.013) Predicted key stage 2 –0.002 –0.038 –0.062 –0.091 0.023** 0.006 (0.026) (0.025) (0.047) (0.052) (0.008) (0.009) Age-7 ks1 tests –0.002 –0.041 –0.033 –0.038* 0.027*** 0.012 (0.032) (0.030) (0.019) (0.017) (0.008) (0.009) FSM 0.016 0.022 0.058* 0.064*** 0.003* 0.005*** (0.013) (0.012) (0.027) (0.019) (0.001) (0.001) Girls 0.003 0.012 –0.000 0.006 0.003 0.002 (0.006) (0.008) (0.006) (0.006) (0.001) (0.001) White –0.019 –0.002 –0.042 –0.039 –0.138*** –0.156*** (0.025) (0.030) (0.023) (0.022) (0.015) (0.018) Age (months) –0.012 0.024 0.013 –0.014 –0.001 –0.011 (0.046) (0.041) (0.040) (0.041) (0.013) (0.010) English first language 0.000 –0.015 –0.020 –0.044 –0.099*** –0.112*** (0.039) (0.039) (0.034) (0.024) (0.016) (0.018) School cohort students –2.395 14.424 –4.831 –6.940 13.750* 19.722*** (17.876) (20.045) (15.532) (11.703) (6.018) (5.855) Ln home-school distance –0.084 –0.128 –0.086 –0.096 0.001 0.005 (0.055) (0.068) (0.049) (0.051) (0.017) (0.016) Live in adjacent LEA –0.036 0.001 –0.040 –0.017 0.019*** 0.033*** (0.024) (0.019) (0.020) (0.016) (0.004) (0.004) School movers in LEA –0.015* –0.007 –0.012 0.000 –0.006** 0.006** (0.007) (0.006) (0.008) (0.005) (0.002) (0.002) School movers out LEA 0.005 0.006 0.011 0.007 0.022*** 0.030*** (0.009) (0.009) (0.010) (0.010) (0.002) (0.002) Movers into school during –0.029* –0.015 –0.004 0.009 –0.003 0.003 ks2 (0.012) (0.019) (0.014) (0.018) (0.003) (0.003) Home house price index –0.007 –0.013 –0.022 –0.014 0.176*** 0.237*** (0.021) (0.031) (0.030) (0.024) (0.017) (0.021) House prices 1 km radius –0.008 –0.001 –0.007 –0.007 0.240*** 0.356*** (0.025) (0.021) (0.027) (0.018) (0.023) (0.026) Home area high quals 0.011 0.011 0.007 0.001 0.077*** 0.084*** (0.009) (0.013) (0.009) (0.013) (0.006) (0.006) Home area no quals 0.004 –0.000 0.007 0.005 –0.051*** –0.070*** (0.008) (0.011) (0.010) (0.009) (0.005) (0.005) Home social tenants 0.035* 0.034 0.050* 0.055** 0.038*** 0.026** (0.017) (0.021) (0.021) (0.020) (0.008) (0.009) Home area born United Kingdom 0.000 0.006 –0.008 –0.006 –0.078*** –0.094*** (0.011) (0.014) (0.008) (0.008) (0.007) (0.008) Home area employed –0.013* –0.009 –0.017 –0.016* 0.009** 0.024*** (0.006) (0.008) (0.009) (0.007) (0.003) (0.003) Home area depriv. index –0.000 0.002 0.023 0.023 0.020*** 0.002 (0.018) (0.012) (0.026) (0.018) (0.004) (0.004) LEA workforce ln pay 0.020 0.060 0.029 0.061 0.090*** 0.125*** (0.042) (0.054) (0.045) (0.054) (0.011) (0.013) Max/min col obs. 9,080/4,245 9,080/4,245 18,408/8,710 18,408/8,710 18,408/8,710 18,408/8,710 (1) (2) (3) (4) (5) (6) Matched on location and FSM Matched on location only Controlling for FSM only Std LEA grant Std ACA Std LEA grant Std ACA Std LEA grant Std ACA Expend. per pupil 0.425*** 0.280*** 0.477*** 0.419*** 0.641*** 0.449*** (0.065) (0.075) (0.083) (0.061) (0.017) (0.030) Key stage 2 0.133** 0.064* 0.034 –0.022 0.096*** 0.085*** (0.045) (0.032) (0.067) (0.066) (0.014) (0.013) Predicted key stage 2 –0.002 –0.038 –0.062 –0.091 0.023** 0.006 (0.026) (0.025) (0.047) (0.052) (0.008) (0.009) Age-7 ks1 tests –0.002 –0.041 –0.033 –0.038* 0.027*** 0.012 (0.032) (0.030) (0.019) (0.017) (0.008) (0.009) FSM 0.016 0.022 0.058* 0.064*** 0.003* 0.005*** (0.013) (0.012) (0.027) (0.019) (0.001) (0.001) Girls 0.003 0.012 –0.000 0.006 0.003 0.002 (0.006) (0.008) (0.006) (0.006) (0.001) (0.001) White –0.019 –0.002 –0.042 –0.039 –0.138*** –0.156*** (0.025) (0.030) (0.023) (0.022) (0.015) (0.018) Age (months) –0.012 0.024 0.013 –0.014 –0.001 –0.011 (0.046) (0.041) (0.040) (0.041) (0.013) (0.010) English first language 0.000 –0.015 –0.020 –0.044 –0.099*** –0.112*** (0.039) (0.039) (0.034) (0.024) (0.016) (0.018) School cohort students –2.395 14.424 –4.831 –6.940 13.750* 19.722*** (17.876) (20.045) (15.532) (11.703) (6.018) (5.855) Ln home-school distance –0.084 –0.128 –0.086 –0.096 0.001 0.005 (0.055) (0.068) (0.049) (0.051) (0.017) (0.016) Live in adjacent LEA –0.036 0.001 –0.040 –0.017 0.019*** 0.033*** (0.024) (0.019) (0.020) (0.016) (0.004) (0.004) School movers in LEA –0.015* –0.007 –0.012 0.000 –0.006** 0.006** (0.007) (0.006) (0.008) (0.005) (0.002) (0.002) School movers out LEA 0.005 0.006 0.011 0.007 0.022*** 0.030*** (0.009) (0.009) (0.010) (0.010) (0.002) (0.002) Movers into school during –0.029* –0.015 –0.004 0.009 –0.003 0.003 ks2 (0.012) (0.019) (0.014) (0.018) (0.003) (0.003) Home house price index –0.007 –0.013 –0.022 –0.014 0.176*** 0.237*** (0.021) (0.031) (0.030) (0.024) (0.017) (0.021) House prices 1 km radius –0.008 –0.001 –0.007 –0.007 0.240*** 0.356*** (0.025) (0.021) (0.027) (0.018) (0.023) (0.026) Home area high quals 0.011 0.011 0.007 0.001 0.077*** 0.084*** (0.009) (0.013) (0.009) (0.013) (0.006) (0.006) Home area no quals 0.004 –0.000 0.007 0.005 –0.051*** –0.070*** (0.008) (0.011) (0.010) (0.009) (0.005) (0.005) Home social tenants 0.035* 0.034 0.050* 0.055** 0.038*** 0.026** (0.017) (0.021) (0.021) (0.020) (0.008) (0.009) Home area born United Kingdom 0.000 0.006 –0.008 –0.006 –0.078*** –0.094*** (0.011) (0.014) (0.008) (0.008) (0.007) (0.008) Home area employed –0.013* –0.009 –0.017 –0.016* 0.009** 0.024*** (0.006) (0.008) (0.009) (0.007) (0.003) (0.003) Home area depriv. index –0.000 0.002 0.023 0.023 0.020*** 0.002 (0.018) (0.012) (0.026) (0.018) (0.004) (0.004) LEA workforce ln pay 0.020 0.060 0.029 0.061 0.090*** 0.125*** (0.042) (0.054) (0.045) (0.054) (0.011) (0.013) Max/min col obs. 9,080/4,245 9,080/4,245 18,408/8,710 18,408/8,710 18,408/8,710 18,408/8,710 Notes: Regressions of row on column variable, aggregated to school-by-year cells, weighted by observations per cell. Standardised coefficients. Specifications (1)–(4) include school-pair-by-year fixed effects. See text for further details. *Significant at 5%; **significant at 1%; ***significant at 0.1%. View Large Table 2. “Balancing” tests on boundary schools. (1) (2) (3) (4) (5) (6) Matched on location and FSM Matched on location only Controlling for FSM only Std LEA grant Std ACA Std LEA grant Std ACA Std LEA grant Std ACA Expend. per pupil 0.425*** 0.280*** 0.477*** 0.419*** 0.641*** 0.449*** (0.065) (0.075) (0.083) (0.061) (0.017) (0.030) Key stage 2 0.133** 0.064* 0.034 –0.022 0.096*** 0.085*** (0.045) (0.032) (0.067) (0.066) (0.014) (0.013) Predicted key stage 2 –0.002 –0.038 –0.062 –0.091 0.023** 0.006 (0.026) (0.025) (0.047) (0.052) (0.008) (0.009) Age-7 ks1 tests –0.002 –0.041 –0.033 –0.038* 0.027*** 0.012 (0.032) (0.030) (0.019) (0.017) (0.008) (0.009) FSM 0.016 0.022 0.058* 0.064*** 0.003* 0.005*** (0.013) (0.012) (0.027) (0.019) (0.001) (0.001) Girls 0.003 0.012 –0.000 0.006 0.003 0.002 (0.006) (0.008) (0.006) (0.006) (0.001) (0.001) White –0.019 –0.002 –0.042 –0.039 –0.138*** –0.156*** (0.025) (0.030) (0.023) (0.022) (0.015) (0.018) Age (months) –0.012 0.024 0.013 –0.014 –0.001 –0.011 (0.046) (0.041) (0.040) (0.041) (0.013) (0.010) English first language 0.000 –0.015 –0.020 –0.044 –0.099*** –0.112*** (0.039) (0.039) (0.034) (0.024) (0.016) (0.018) School cohort students –2.395 14.424 –4.831 –6.940 13.750* 19.722*** (17.876) (20.045) (15.532) (11.703) (6.018) (5.855) Ln home-school distance –0.084 –0.128 –0.086 –0.096 0.001 0.005 (0.055) (0.068) (0.049) (0.051) (0.017) (0.016) Live in adjacent LEA –0.036 0.001 –0.040 –0.017 0.019*** 0.033*** (0.024) (0.019) (0.020) (0.016) (0.004) (0.004) School movers in LEA –0.015* –0.007 –0.012 0.000 –0.006** 0.006** (0.007) (0.006) (0.008) (0.005) (0.002) (0.002) School movers out LEA 0.005 0.006 0.011 0.007 0.022*** 0.030*** (0.009) (0.009) (0.010) (0.010) (0.002) (0.002) Movers into school during –0.029* –0.015 –0.004 0.009 –0.003 0.003 ks2 (0.012) (0.019) (0.014) (0.018) (0.003) (0.003) Home house price index –0.007 –0.013 –0.022 –0.014 0.176*** 0.237*** (0.021) (0.031) (0.030) (0.024) (0.017) (0.021) House prices 1 km radius –0.008 –0.001 –0.007 –0.007 0.240*** 0.356*** (0.025) (0.021) (0.027) (0.018) (0.023) (0.026) Home area high quals 0.011 0.011 0.007 0.001 0.077*** 0.084*** (0.009) (0.013) (0.009) (0.013) (0.006) (0.006) Home area no quals 0.004 –0.000 0.007 0.005 –0.051*** –0.070*** (0.008) (0.011) (0.010) (0.009) (0.005) (0.005) Home social tenants 0.035* 0.034 0.050* 0.055** 0.038*** 0.026** (0.017) (0.021) (0.021) (0.020) (0.008) (0.009) Home area born United Kingdom 0.000 0.006 –0.008 –0.006 –0.078*** –0.094*** (0.011) (0.014) (0.008) (0.008) (0.007) (0.008) Home area employed –0.013* –0.009 –0.017 –0.016* 0.009** 0.024*** (0.006) (0.008) (0.009) (0.007) (0.003) (0.003) Home area depriv. index –0.000 0.002 0.023 0.023 0.020*** 0.002 (0.018) (0.012) (0.026) (0.018) (0.004) (0.004) LEA workforce ln pay 0.020 0.060 0.029 0.061 0.090*** 0.125*** (0.042) (0.054) (0.045) (0.054) (0.011) (0.013) Max/min col obs. 9,080/4,245 9,080/4,245 18,408/8,710 18,408/8,710 18,408/8,710 18,408/8,710 (1) (2) (3) (4) (5) (6) Matched on location and FSM Matched on location only Controlling for FSM only Std LEA grant Std ACA Std LEA grant Std ACA Std LEA grant Std ACA Expend. per pupil 0.425*** 0.280*** 0.477*** 0.419*** 0.641*** 0.449*** (0.065) (0.075) (0.083) (0.061) (0.017) (0.030) Key stage 2 0.133** 0.064* 0.034 –0.022 0.096*** 0.085*** (0.045) (0.032) (0.067) (0.066) (0.014) (0.013) Predicted key stage 2 –0.002 –0.038 –0.062 –0.091 0.023** 0.006 (0.026) (0.025) (0.047) (0.052) (0.008) (0.009) Age-7 ks1 tests –0.002 –0.041 –0.033 –0.038* 0.027*** 0.012 (0.032) (0.030) (0.019) (0.017) (0.008) (0.009) FSM 0.016 0.022 0.058* 0.064*** 0.003* 0.005*** (0.013) (0.012) (0.027) (0.019) (0.001) (0.001) Girls 0.003 0.012 –0.000 0.006 0.003 0.002 (0.006) (0.008) (0.006) (0.006) (0.001) (0.001) White –0.019 –0.002 –0.042 –0.039 –0.138*** –0.156*** (0.025) (0.030) (0.023) (0.022) (0.015) (0.018) Age (months) –0.012 0.024 0.013 –0.014 –0.001 –0.011 (0.046) (0.041) (0.040) (0.041) (0.013) (0.010) English first language 0.000 –0.015 –0.020 –0.044 –0.099*** –0.112*** (0.039) (0.039) (0.034) (0.024) (0.016) (0.018) School cohort students –2.395 14.424 –4.831 –6.940 13.750* 19.722*** (17.876) (20.045) (15.532) (11.703) (6.018) (5.855) Ln home-school distance –0.084 –0.128 –0.086 –0.096 0.001 0.005 (0.055) (0.068) (0.049) (0.051) (0.017) (0.016) Live in adjacent LEA –0.036 0.001 –0.040 –0.017 0.019*** 0.033*** (0.024) (0.019) (0.020) (0.016) (0.004) (0.004) School movers in LEA –0.015* –0.007 –0.012 0.000 –0.006** 0.006** (0.007) (0.006) (0.008) (0.005) (0.002) (0.002) School movers out LEA 0.005 0.006 0.011 0.007 0.022*** 0.030*** (0.009) (0.009) (0.010) (0.010) (0.002) (0.002) Movers into school during –0.029* –0.015 –0.004 0.009 –0.003 0.003 ks2 (0.012) (0.019) (0.014) (0.018) (0.003) (0.003) Home house price index –0.007 –0.013 –0.022 –0.014 0.176*** 0.237*** (0.021) (0.031) (0.030) (0.024) (0.017) (0.021) House prices 1 km radius –0.008 –0.001 –0.007 –0.007 0.240*** 0.356*** (0.025) (0.021) (0.027) (0.018) (0.023) (0.026) Home area high quals 0.011 0.011 0.007 0.001 0.077*** 0.084*** (0.009) (0.013) (0.009) (0.013) (0.006) (0.006) Home area no quals 0.004 –0.000 0.007 0.005 –0.051*** –0.070*** (0.008) (0.011) (0.010) (0.009) (0.005) (0.005) Home social tenants 0.035* 0.034 0.050* 0.055** 0.038*** 0.026** (0.017) (0.021) (0.021) (0.020) (0.008) (0.009) Home area born United Kingdom 0.000 0.006 –0.008 –0.006 –0.078*** –0.094*** (0.011) (0.014) (0.008) (0.008) (0.007) (0.008) Home area employed –0.013* –0.009 –0.017 –0.016* 0.009** 0.024*** (0.006) (0.008) (0.009) (0.007) (0.003) (0.003) Home area depriv. index –0.000 0.002 0.023 0.023 0.020*** 0.002 (0.018) (0.012) (0.026) (0.018) (0.004) (0.004) LEA workforce ln pay 0.020 0.060 0.029 0.061 0.090*** 0.125*** (0.042) (0.054) (0.045) (0.054) (0.011) (0.013) Max/min col obs. 9,080/4,245 9,080/4,245 18,408/8,710 18,408/8,710 18,408/8,710 18,408/8,710 Notes: Regressions of row on column variable, aggregated to school-by-year cells, weighted by observations per cell. Standardised coefficients. Specifications (1)–(4) include school-pair-by-year fixed effects. See text for further details. *Significant at 5%; **significant at 1%; ***significant at 0.1%. View Large Firstly, in rows (1) and (2), to preview to our main results and provide benchmarks for the balancing tests, we show the relationship between school expenditure and the instruments (analogous to the first stage regressions in the student-level IV analysis that follows in Section 5.3) and the relationship between the instruments and school average test scores at ks2 (analogous to the reduced form in Section 5.3). The first row shows, unsurprisingly, that LEA grant and ACA differentials do affect school expenditure differentials. Row (2) shows that age 11 test scores increase significantly with the LEA grant per pupil and the ACA index in our preferred specification. The coefficient of 0.133 in column (1) row (2) implies that a £1,000 per pupil per year difference in LEA grant each year over primary school because ks1 leads to a 0.31 standard deviation (=1000/435 × 0.133) increase in ks2 scores. In the other cells in columns (1) and (2), we change the dependent variable to one of a number of variables describing the students, the school, the residential area in which each student lives (Census Output Area), and the mean basic pay for workers in the Local Education Authority.19 The aim here is to look for differences in these salient characteristics between school locations in LEAs with high and low levels of funding and for sorting of students between high- and low-funded LEAs. Row (3) reports balancing on predicted ks2 scores using a linear index of the basic predetermined pupil characteristics (prior ks1 achievement at age 7, FSM, gender, age, ethnicity, English first language). This index is estimated from a within-school regression of student ks2 test scores on these characteristics. As shown in Appendix A, we effectively eliminate selection on this index by our school-matching procedure. In the remaining columns, we investigate balancing on the individual components of this index and on a range of other covariates. From the remaining rows of columns (1) and (2) in Table 2, it is evident that most coefficients are statistically insignificant at the 5% level and relatively small in magnitude. The only exceptions are that mobility into high-LEA grant schools is lower than in low-LEA grant schools, and pupils in high-funded schools seem to come from residential areas with more social tenants and lower employment, though these coefficients too are insignificant once we look at balancing on the ACA index. Note to convincingly reject the null of no relationship between the funding variables and these 21 characteristics would require a p-value of 0.24% (=0.05/21) and a t-statistic of over 3. Particularly important is the lack of significance and small magnitude of the coefficient on earlier test scores of the students at age 7 (row 5). Evidently, students in schools with higher LEA grants were not performing significantly better much earlier in their education, which reinforces the argument that our findings are not driven by sorting. Equally important is the lack of any evidence that house prices are higher, either in the student's home residential area, or in the 1 km radius around the school (rows 16 and 17). This lack of response of housing prices to school funding in England is consistent with Gibbons, Machin and Silva (2013) who also report no response of housing prices to expenditure despite evidence of a price response to school peer groups and value-added. Whatever components of school performance and inputs households are paying for, expenditure appears not to be one of them. As discussed in Section 4, this is not as surprising as it might first appear given that higher expenditure and/or LEA grant send mixed signals about school demographics and performance due to the compensatory funding mechanisms (as evidenced by the strong negative raw correlation between expenditure and pupil performance). In short, parents would need a lot of information and analysis to deduce whether to choose the higher or lower expenditure school if they are in search of both better outcomes for their child and other desirable school attributes like good peer groups, especially because existing evidence on the benefits of school expenditure is equivocal. Columns (3)–(6) illustrate why we need to match neighbouring schools in similar parts of the FSM distribution. If we do not match on FSM but match only on distance when forming school pairs, the coefficients in the majority of these balancing tests from row (3) onwards remain small and nonsignificant and the pattern is broadly similar to that in column (1). However, the coefficient on FSM is substantial and highly significant. This is because, as discussed in Section 4.3, FSM is a school-level proxy for the LEA-level additional needs index that enters directly into the LEA grant. Consequently, FSM is also highly correlated with the LEA-level ACA index, because the additional needs and ACA indices are highly correlated at LEA level (inner-urban LEAs with high wages also have high levels of poverty). Now, there is no evidence of an effect of resources on ks2 test scores (row 2) because the proportion of FSM students is a crucial confounder that is strongly negatively associated with school ks2 performance. Comparing nearest neighbour schools and controlling for spatial trends alone is insufficient to control for FSM differences between schools (in the index vs in Section 4). Conversely, if we control for FSM but do not compare nearest-neighbour schools in columns (5) and (6), we do find a positive association between LEA grants or the ACA index and school performance. However, balancing on the other characteristics is extremely poor, so we could have no confidence that that this estimate is causal. In short, Table 2 provides evidence of the “conditional ignorability” identification condition. We must compare nearest-neighbour schools matched by their FSM proportions in order to successfully eliminate the confounders that are correlated with both LEA grants (mainly through the compensatory funding formula) and school ks2 test scores. The LEA grant instrument performs almost as well as the ACA instrument in this respect. Appendix B presents further evidence of balancing across LEA boundaries, showing graphs typical of standard RDD analysis (though here we are averaging over multiple boundaries, so the patterns should not be interpreted in the same way as a standard RDD where there is only one). 5.3. Regression Results Table 3 presents our main regression results on the relationship between school resources and ks2 scores in our sample of boundary schools. The coefficients are scaled to show the change in standardised student ks2 (age-11) test scores for a £1,000 of additional per-student-per-year expenditure, where expenditure is a moving average over the preceding years (up to 4) before the tests. Standard errors are adjusted for clustering on LEA boundaries to correct for the correlation between observations induced by our sample structure, plus arbitrary spatial correlation along the LEA boundary. Table 3. Main results on the effects of expenditure on student ks2 test scores at age 11. (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) School expenditure OLS School expenditure OLS OLS school expenditure, school group fixed fx OLS school expenditure, school group fixed fx Reduced form sharp discontinuity: LEA grant per pupil Reduced form sharp discontinuity: LEA grant per pupil IV fuzzy discontinuity: LEA grant instrument IV fuzzy discontinuity: LEA grant instrument IV fuzzy discontinuity: ACA index instrument IV fuzzy discontinuity: ACA index instrument Total money per pupil –0.156*** 0.012 –0.049* 0.044 0.287*** 0.282*** 0.362*** 0.360*** 0.265* 0.365*** (annual mean, £1000s) (0.018) (0.011) (0.023) (0.026) (0.069) (0.064) (0.086) (0.074) (0.111) (0.097) Control variables No Yes No Yes No Yes No Yes No Yes Distance to boundary cubic No No Yes Yes Yes Yes Yes Yes Yes Yes First stage: F-stat – – – – – 85.32 107.0 28.21 54.40 First stage: Coefficient – – – – – 0.792 0.782 2.228 2.432 (0.086) (0.076) (0.419) (0.330) Observations 379,194 379,194 379,194 379,194 379,194 379,194 379,194 379,194 379,194 379,194 School-pair-by-year fixed fx – – 4,540 4,540 4,540 4,540 4,540 4,540 4,540 4,540 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) School expenditure OLS School expenditure OLS OLS school expenditure, school group fixed fx OLS school expenditure, school group fixed fx Reduced form sharp discontinuity: LEA grant per pupil Reduced form sharp discontinuity: LEA grant per pupil IV fuzzy discontinuity: LEA grant instrument IV fuzzy discontinuity: LEA grant instrument IV fuzzy discontinuity: ACA index instrument IV fuzzy discontinuity: ACA index instrument Total money per pupil –0.156*** 0.012 –0.049* 0.044 0.287*** 0.282*** 0.362*** 0.360*** 0.265* 0.365*** (annual mean, £1000s) (0.018) (0.011) (0.023) (0.026) (0.069) (0.064) (0.086) (0.074) (0.111) (0.097) Control variables No Yes No Yes No Yes No Yes No Yes Distance to boundary cubic No No Yes Yes Yes Yes Yes Yes Yes Yes First stage: F-stat – – – – – 85.32 107.0 28.21 54.40 First stage: Coefficient – – – – – 0.792 0.782 2.228 2.432 (0.086) (0.076) (0.419) (0.330) Observations 379,194 379,194 379,194 379,194 379,194 379,194 379,194 379,194 379,194 379,194 School-pair-by-year fixed fx – – 4,540 4,540 4,540 4,540 4,540 4,540 4,540 4,540 Notes: Table reports regression coefficients and standard errors from student-level regressions, 2004–2009. Estimation based on all 840 Community schools in England that can be matched by location and FSM intake (the main determinant of within-LEA funding differences). Schools are matched to others that are within 2 km in adjacent LEA and with FSM proportion within 10 percentiles (of the national distribution of boundary schools). Regressions in columns (3)–(10) include school pair-by-year fixed effects that is, estimation is based on differences between matched pairs. Dependent variable is standardised mean student total score in English, Maths, and Science. “Distance to boundary cubic” means control variables for third-order polynomial series in distance from school to LEA boundary interacted with high-/low-funded side of boundary (based on LEA grant). Control variables are as follows: student key stage 1 (age 7) test scores (15 dummies), gender, FSM, month of birth, English first language, nine ethnic dummies, national funding formula educational needs index, number of pupils in school year group. Standard errors robust to heteroscedasticity and autocorrelation along LEA-pair boundary. “LEA grant instrument” is mean grant per pupil from LEA (mean across primary schools in LEA as a whole); ACA index instrument is LEA-specific ACA in central government funding formula (see text) *Significant at 5%; ***significant at 0.1%. View Large Table 3. Main results on the effects of expenditure on student ks2 test scores at age 11. (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) School expenditure OLS School expenditure OLS OLS school expenditure, school group fixed fx OLS school expenditure, school group fixed fx Reduced form sharp discontinuity: LEA grant per pupil Reduced form sharp discontinuity: LEA grant per pupil IV fuzzy discontinuity: LEA grant instrument IV fuzzy discontinuity: LEA grant instrument IV fuzzy discontinuity: ACA index instrument IV fuzzy discontinuity: ACA index instrument Total money per pupil –0.156*** 0.012 –0.049* 0.044 0.287*** 0.282*** 0.362*** 0.360*** 0.265* 0.365*** (annual mean, £1000s) (0.018) (0.011) (0.023) (0.026) (0.069) (0.064) (0.086) (0.074) (0.111) (0.097) Control variables No Yes No Yes No Yes No Yes No Yes Distance to boundary cubic No No Yes Yes Yes Yes Yes Yes Yes Yes First stage: F-stat – – – – – 85.32 107.0 28.21 54.40 First stage: Coefficient – – – – – 0.792 0.782 2.228 2.432 (0.086) (0.076) (0.419) (0.330) Observations 379,194 379,194 379,194 379,194 379,194 379,194 379,194 379,194 379,194 379,194 School-pair-by-year fixed fx – – 4,540 4,540 4,540 4,540 4,540 4,540 4,540 4,540 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) School expenditure OLS School expenditure OLS OLS school expenditure, school group fixed fx OLS school expenditure, school group fixed fx Reduced form sharp discontinuity: LEA grant per pupil Reduced form sharp discontinuity: LEA grant per pupil IV fuzzy discontinuity: LEA grant instrument IV fuzzy discontinuity: LEA grant instrument IV fuzzy discontinuity: ACA index instrument IV fuzzy discontinuity: ACA index instrument Total money per pupil –0.156*** 0.012 –0.049* 0.044 0.287*** 0.282*** 0.362*** 0.360*** 0.265* 0.365*** (annual mean, £1000s) (0.018) (0.011) (0.023) (0.026) (0.069) (0.064) (0.086) (0.074) (0.111) (0.097) Control variables No Yes No Yes No Yes No Yes No Yes Distance to boundary cubic No No Yes Yes Yes Yes Yes Yes Yes Yes First stage: F-stat – – – – – 85.32 107.0 28.21 54.40 First stage: Coefficient – – – – – 0.792 0.782 2.228 2.432 (0.086) (0.076) (0.419) (0.330) Observations 379,194 379,194 379,194 379,194 379,194 379,194 379,194 379,194 379,194 379,194 School-pair-by-year fixed fx – – 4,540 4,540 4,540 4,540 4,540 4,540 4,540 4,540 Notes: Table reports regression coefficients and standard errors from student-level regressions, 2004–2009. Estimation based on all 840 Community schools in England that can be matched by location and FSM intake (the main determinant of within-LEA funding differences). Schools are matched to others that are within 2 km in adjacent LEA and with FSM proportion within 10 percentiles (of the national distribution of boundary schools). Regressions in columns (3)–(10) include school pair-by-year fixed effects that is, estimation is based on differences between matched pairs. Dependent variable is standardised mean student total score in English, Maths, and Science. “Distance to boundary cubic” means control variables for third-order polynomial series in distance from school to LEA boundary interacted with high-/low-funded side of boundary (based on LEA grant). Control variables are as follows: student key stage 1 (age 7) test scores (15 dummies), gender, FSM, month of birth, English first language, nine ethnic dummies, national funding formula educational needs index, number of pupils in school year group. Standard errors robust to heteroscedasticity and autocorrelation along LEA-pair boundary. “LEA grant instrument” is mean grant per pupil from LEA (mean across primary schools in LEA as a whole); ACA index instrument is LEA-specific ACA in central government funding formula (see text) *Significant at 5%; ***significant at 0.1%. View Large As discussed in Section 4, we implement the cross–LEA-boundary differencing design on student-level data, with each school s paired with a matched nearest-neighbour s’ in an adjacent LEA. Columns (1) and (2) present a basic ordinary least squares (OLS) regression of test scores on school expenditure for comparison purposes (without differencing across the boundary). In columns (3)–(10), we then estimate effects using variation in cross-boundary differences in expenditure, controlling for school-pair-by-year fixed effects using the within-groups estimator. To ensure identification comes from differences in expenditure and test scores close to the boundary, we also control for the spatial assignment variable using distance-to-boundary polynomials interacted with a dummy indicating the high-funded side of the boundary (based on the average LEA grant). The first column in each column pair shows results without any control variables other than these distance-to-boundary controls. The second column in each pair includes additional control variables for student characteristics (FSM, ethnic group dummies, gender, month of birth, English first language, prior, age-7 ks1 test scores, school enrolment). Looking at the OLS results in column (1), we see a strong negative association between school expenditure and student test scores. This association arises due to the needs-based resource allocation to schools and cannot be interpreted as causal. Column (2) adds in the control variable set, which drives the coefficient towards zero (and insignificance) because these variables at least partially control for factors that jointly determine resource allocation and student achievement. The results in column (2) are typical of the low or insignificant coefficients that come out of OLS regressions in the literature that try to deal with the endogeneity of school resources by controlling for observable characteristics (see the literature documented by Hanushek 2003). The second set of results (columns 3 and 4) also uses school-specific expenditures as the resource variable, but the regression controls for school-pair-by-year fixed effects, and the distance-to-boundary controls, that is, the effect of school expenditure is estimated using differences between the schools in the pairs matched across the LEA boundaries. The estimates in both columns are again small and statistically insignificant, though switch sign when the control variables are added. In this case, the expenditure differences between matched schools on different sides of the boundary occur due to a combination of differences in the LEA grant between the adjacent LEAs and idiosyncratic variation in expenditure between the schools discussed in Sections 4.2 and 5.1. Again, these estimates cannot be interpreted as causal because the idiosyncratic, non-LEA based, sources of funding are not systematically related to geographical location nor FSM, so are not controlled by the spatial differencing and matching design. They will depend on factors such as random variation in school intake characteristics that attract additional funding (e.g., how many children are diagnosed with SEN and the proportion of ethnic minorities). Given the comparable variances of the school-specific and LEA grant components of income documented in Table 1, it is unsurprising that any causal impact from the LEA grant differences is offset by downward biases coming from other compensatory school-specific income sources. These results demonstrate that spatial differencing between closely spaced schools and controlling flexibly for spatial location is not on its own an effective strategy to deal with the endogeneity of school-level expenditures. To overcome this problem, columns (5) and (6) implement the sharp, reduced form discontinuity design from equations (2) and (3) in which the expenditure treatment variable is the LEA-level grant rather than school-specific, level of funding per pupil per year.20 Now, differences in funding between each of the schools in a pair are due solely to each school being on different sides of the boundary and hence exposed to a different level of LEA grant. With this set up, we need only eliminate confounders that are correlated with the LEA grant, which is determined simply by the national formula. Differencing between the FSM-matched nearest-neighbour schools and including distance-to-boundary polynomials effectively controls for these confounders. These results are not, however, dependent on the specific functional form of the distance controls, as we show later in Section 5.4. When we implement this reduced form, sharp discontinuity design, the coefficients for the effect of funding on ks2 test scores are positive and statistically significant. The coefficients remain relatively stable when we include the extended control variable set—including prior achievement at age 7—in column (6). The effect sizes are large, implying that a £1,000 increase in mean LEA per pupil per year expenditure during the ks2 period increases student performance at ks2 by 0.29 standard deviations. The IV spatial discontinuity design estimates are shown in columns (7)–(10). In columns (7)–(8), we use school-specific expenditure per pupil as the treatment variable (as in columns 3–4), but instrument this with the LEA-specific grant per pupil (the treatment variable in columns 5–6) using two-stage least squares. The first-stage coefficients are around 0.8 and the F-statistics are high (85–107), which is not surprising given that the LEA grant is, by construction, a key input into the school budget. The point estimate without the extended control variables set in column (5) is larger than the reduced-form estimates at around 0.36, and this is insensitive to inclusion of the extended control variable set. Columns (9)–(10) present the IV discontinuity estimates using our alternative instrument, the ACA component of the fu